A change of basis is a method to convert vector coordinates with respect to one basis to coordinates with respect to another basis.

Consider a vector ** u** with reference to orthogonal unit basis vectors

*x*,

_{1}*x*

_{2 }and

*x*.

_{3}where *p*, *q* and *r* are the scalar components of the unit basis vectors *x _{1}*,

*x*

_{2 }and

*x*respectively.

_{3}Clearly,

If we define ** u** with respect to another orthogonal reference frame with unit basis vectors

*x’*,

_{1}*x’*

_{2 }and

*x’*(see above diagram), we have:

_{3}and

To find the relationship between the two reference frames, we substitute eq1 in eq4 to give

Since the dot product of two vectors gives a scalar, we can write eq5, eq6 and eq7 as

which can be expressed in the following matrix equation:

where .

The matrix containing *a _{ij}* is known as the

*. We can rewrite eq8 as:*

**change of basis matrix**where and .

Eq9 is often written as a short form by omitting the summation symbol:

For example, the change of basis from the orthogonal basis *x _{j}* to the orthogonal basis

*x’*as a rotation about the

_{i }*x*-axis, where

_{3}*x*coincides with

_{3 }*x’*(

_{3}*x’*=

_{3 }*x*and is perpendicular to the plane of the page), is depicted as:

_{3}To determine the change of basis matrix, we have

Therefore,

Consider again the vector ** u **with reference to a coordinate system that is defined by the orthogonal unit basis vectors

*x*,

_{1}*x*

_{2 }and

*x*(see diagram below).

_{3}The rotation of ** u **with respect to this fixed coordinate system is expressed as:

Compared to the change of basis of ** u **that was described earlier, the rotation matrix

*R*is the inverse of the change of basis matrix (similarly, a reflection matrix is the inverse of the change of basis matrix by a reflection). In other words, the rotation of

**by**

*u**θ*can be perceived a change of basis, with the coordinate system rotated by –

*θ*. This implies that we can analyse a problem in two ways.